I realise that something would need to be constantly on listening, but it wouldn't have to be very sensitive, it would be listening for gunshots and would need to record the sound, GPS location and anything else that could be used to help in locating the sound.

Loud is relative. dropping some change into your pocket with your phone would register as being far louder than a gunshot outside your house while you're inside eating breakfast. There is an area, somewhere in the states, with a system of what look kind of like Dr. Suess lamp posts, with a bunch of cones coming off them in different directions, and then a camera. It can use it's array of microphones (the cones) to locate the source of the sound, and it swings the camera in that direction. The analysis of the sound to determine if it was or was not a gunshot sounded computationally expensive, and the microphones used in that system were far better suited to the task than the ones found in cell phones.

I think for the system you're thinking of to be useful, there would need to be a way to tell a loud car horn from a gunshot or the system would register millions of false positives for every letitimate "Bad sound".